308 
sional character of space. Were space a 
form of thought, so would be time and mo- 
tion, and kinematics would also be a part 
of pure mathematics. So he relegates 
geometry to the real sciences, and he has a 
difficulty in retaining arithmetic even, for 
is it not based on axioms, whereas a formal 
science is based on conventions ? 
From the notion of the combination of 
terms he deduces that the placing of the 
brackets and the order of the terms may or 
may not be indifferent. There is a syn- 
thetic combination and an analytic combi- 
nation; when the latter is unambiguous 
(that is) a—a=0O) then the placing of 
the brackets and the order of the terms is 
indifferent ; synthetic combination is then 
called addition, and the analytic subtrac- 
tion. Thus in Grassmann’s view the com- 
mutative and associative laws are involved 
in the ideas of addition and subtraction. 
It may be observed that the old difficulty 
with subtraction is due to the fact that it is 
not thoroughly commutative, and that it is 
only to the generalized idea of composition 
that the commutative law applies. Be- 
sides, to define addition so as to exclude 
terms having a real order is an arbitrary 
restriction of algebra. 
According to Grassmann’s view multipli- 
cation is a combination of a higher order; 
that is, he assumes as the definition of mul- 
tiplication the distributive principle in the 
two-fold form 
(a+ b)c=ac+ beandc(a+b)=ca+ cb. 
It.may be observed, however, that the true 
expression for the distributive principle is 
(a +b) (e+ d) = ace + ad + be + bd, 
which assumes that if there is any real order 
of the terms there can be only one real order 
abed. Y 
As regards the laws of indices he says 
that involution is a combination of the 
the third order, and that for the sake of 
shortness he will omit all consideration of 
SCIENCE. 
[N.S. Vox. X. No. 246 
it. Besides, its formal definition would be 
of no use, for in the nature of things it can 
be applied only in the special sciences 
through real definitions. This failure to 
treat of the index laws tells against his 
whole theory of the nature of algebra. In 
fact, these laws are the touchstone where- 
by the soundness of any theory of the 
foundations of algebra may be tested. 
In 1867 Hermann Hankel published his 
‘Theory of Complex Numbers.’ The full 
title of the work is‘ Theorie der complexen 
Zahiensysteme insbesondere der gemeinen imar- 
gindren Zahlen wnd der Hamilton’schen Quater- 
nionen nebst threr geometrischen Darstellung.’ 
He had studied the writings of both Hamil- 
ton and Grassmann, and the aim of the book 
is to give a complete theory of the several 
systems, uniting them all under the notion 
of complex number. From the title we 
gather that he considered the algebraic 
imaginaries and the Hamiltonian quater- 
nions as two distinct systems, formal in 
their nature, but having a representation 
in space. He begins with positive integer 
numbers, and finds from a consideration of 
the notion that the addition of such num- 
bers satisfies the two laws of association 
and commutation, which he treats as inde- 
pendent of one another. But as regards 
the notion of the multiplication of such 
numbers he says that the truth of the 
commutative law or of the associative law 
is not self-evident ; that the former law can 
be proved by a geometric construction in a 
plane, and the latter by a geometric con- 
struction in space. As regards the dis- 
tributive law he says merely that it is a 
universal property of multiplication. As 
regards the base and index relation he 
says that neither the commutative law 
nor the associative law applies; he enun- 
ciates the same three index laws as De 
Morgan, but does not say whether they 
are self-evident or require a proof by geo- 
metric construction. Here, then, in a pro- 
